EXISTENCE AND RELAXATION OF BV SOLUTIONS FOR A SWEEPING PROCESS WITH PROX-REGULAR SETS

In this paper, we study a discontinuous sweeping process involving prox-regular sets and a multivalued perturbation in a separable Hilbert space. The variation of the moving set is controlled by a positive Radon measure and the values of the perturbation are closed, not necessarily convex sets. A solution of the sweeping process is a pair consisting of a right continuous function of bounded variation whose differential measure is absolutely continuous with respect to some positive Radon measure and an integrable selector of the perturbation considered on this function. Along with the original perturbation, we consider the perturbation with the convexified values. We prove theorems of existence and relaxation of solutions. The latter means the density of the solution set of the sweeping process with the original perturbation in the solution set of the sweeping process with the convexified perturbation. These solution sets are considered as subsets of the Cartesian product of the space of right continuous regular functions and the space of integrable functions. These spaces are endowed with the topology of uniform convergence on an interval and the weak topology, respectively. It is shown that the solution set of the sweeping process with the convexified perturbation is compact in the aforementioned space. The results we obtain can be applied to study optimal control problems with both nonconvex and convexified control constraints and to establish connections between solutions to these problems.